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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 3 Ver. VI (May. - Jun. 2016), PP 66-85
www.iosrjournals.org
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 66 | Page
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past
An Infinite Moving Plate Embedded In A Porous Medium With
Thermal Radiation Temperature Dependent Heat Source And
Variable Suction
1D.Praveena
1S.Vijayakumar Varma
2B. Mamatha
1Department of Mathematics, S.V.University, Tirupati-517 502, A.P. India
2Department of Mathematics, S.P.W. Degree and P.G.College, Tirupati – 517 502, A.P.
Abstract: This chapter deals with the MHD convective heat transfer flow of kuvshinski fluid through a porous
medium past an infinite moving porous plate in the presence of temperature dependent heat source radiation
and variable suction. The fluid flow direction is taken as x-axis and normal to it as y-axis. A uniform magnetic
field is applied perpendicular to the flow. The governing non-dimensional equations are analytically when i)
The plate is with constant temperature (CWT) and ii) Theplate is variable temperature (VWT). The expressions
for velocity, temperature fields are obtained. Skin friction coefficient and the rate of heat transfer in terms of
nusselt number (Nu) are also derived. Results are discussed and analysed for various material parameters
through graphs and tables.
Keywords: Kuvshinski fluid; MHD; boundary layer; porous medium; thermal radiation; heat generation.
I. Introduction
The study of radiation in thermal engineering is of great interest for industry point of view,Many
processes in thermal engineering areas occur at high temperature and radiative heat transfer becomes very
important for design of pertinent equipment. The study of flow and heat transfer for an electrically conducting
fluid past a porous plate under the influence of magnetic field has attraction in the interest of many investigators
in view of its applications in many engineering problems such as magnetohydrodynamics generator, plasma
studies, nuclear reactors, geothermal energy extractions and the boundary layer control in the field of
aerodynamics. In In recent times chemical reaction and radiation absorption influences the fluid flow attracted
the attention of engineers and scientists. There are many interesting aspects and analytical solutions to such
problems of flow have been presented by many authors [1] Pohlhausen et al. studied the free convection flow
past a semi – infinite vertical plate by the momentum integral method because free convection flows have wide
applications in industry. [2] Ostrach S solved the non-liner coupled ordinary differential equations numerically
on a computer. [3] P.G. Saffirman studied the stability of laminar flow of a dusty gas. [4] Gill et al. observed
diffusion and heat transfer in laminar free convection boundary layers on a vertical plate. [5] Takhar et al.
studied radiation effects on MHD free convection flow of a radiating gass past a semi-infinite plate.
[6]Soundalker studied approximate solutions for the two diffusional flow of an impressible viscous fluid past an
infinite porous vertical plate with constant suction velocity normal the plate. [7] V.M.Soundalger et al. studied
magnetohydrodynamics (MHD) generator, plasma nuclear reactors, geothermal energy extractions and the
boundary layer control in the field of aerodynamics. [8] Raptis and Kafousians studied influence of a magnetic
field upon the steady free convection flow through a porous medium bounded by an infinite vertical plate with
constant suction velocity. [9].A.A.Raptisstudied Time-varying two-dimensional natural convective heat transfer
on an incompressible, electrically-conductive viscous fluid via a highly porous medium bounded by an infinite
vertical porous plate. [10].H.T.Chenand C.K.Chen observed free convection of non-newtonian Fluid Flow
along a vertical plate embedded in a porous medium. [11] KeeSoohan et al. analysed heat transfer in a pipe
carrying two – phase gas particle suspension. [12]Sharma PR and PankajMathur studied steady laminar free
convection flow of an electrically conducting fluid along a porous hot vertical plate in the presence of heat
source/sink. [13]Takhar H.S et al. observed radiation effects on MHD free convection flow of a radiating gas
past a semi-infinite vertical plate. [14] M.A.Mansour observed forced convection-radiation interaction heat
transfer in boundary layer over a flat plate submerged in a porous medium.[15] Helmy A.K studied MHD
unsteady free convection flows past a vertical porous plate.[16] Edmundo M. et al. analysed Numerical model
for radiative heat transfer analysis in arbitrary shaped axisymmetric enclosures with gases media. [17] A. Raptis
studied radiation and free convection flow through a porous medium.[18] M.A.Hossain et al. observed the effect
of radiation in free convection from a porous vertical plate.[19] M.A. El.Hakiem considered MHD oscillatory
flow on free convection-radiation through a porous medium with constant suction velocity. [20] Acharya M.et
al. observed magnetic field effects on the free convection and mass transfer flow through porous medium with
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 67 | Page
constant suction and constant heat flux. [21] Kim Youn J. studied unsteady MHD convective heat transfer past
a semi-infinite vertical porous moving plate with variable suction. [22] E.M.AboEldahab and M.S.E.Gendy
observed radiation effects on convective heat transfer in an electrically conducting fluid at a stretching surface
with variable viscosity and uniform free stream.[23] E.M.AboEldahab and M.S.E.Gendy studied convective
heat transfer past a continuously moving plate embedded in a non-darcian porous medium in the presence of a
magnetic field. [24] Sahoo P.K et al. studied magnetohydrodynamic unsteady free convection flow past an
infinite vertical plate with constant suction and heat sink. [25] Isreal-cookey et al. observed Influence of viscous
dissipation and radiation on unsteady MHD free –convection flow past an infinite heated vertical plate in a
porous medium with time dependent suction. [26] Raptis et al. observed effect of thermal radiation on MHD
flow.[27] R.K.Varshney and Ram Prakash studied the effects of MHD free convection flow of a visco-elastic
dusty gas through a porous medium induced by the motion of a semi-infinite flat plate moving with velocity
decreasing exponentially with time. [28] R. Muthukumaraswamy and G. Kumar Senthil, observed the heat and
Mass transfer effect on moving vertical plate in the presence of thermal radiation. [29]Chamkaali J. studied
unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat
absorption. [30] S.Shateyi et al. observed magnetohyderodynamics flow past a vertical plate with radiative heat
transfer. [31] Ibrahim et al. studied effect of the chemical reaction and radiation absorption on the unsteady
MHD free convection flow past a semi-infinite vertical permeable moving plate with het source and suction.[32]
P.M Patil, andP.S.Kulkarni studied effects of chemical reaction on free convection flow of a polar fluid through
a porous medium in the presence of Internal heat generation. [33] O.A. Beg et al. observed magneto
hydrodynamic convection flow from a sphere to a non-darcian porous medium with heat generation or
absorption effects network simulation. [34] Kumar H. studied radiativeheat transfer with hydromagenatic flow
and viscous dissipation over a stretching surface in the presence of variable heat flux. [35] O.D.Makinde and T.
Chinyoka, studied numerical investigation of transient heat transfer to hydromagnetic channel flow
withradiative heat and convective cooling. [36] J. Gireesh Kumar, P.V.Satynarayanaobserved Mass transfer
effect on MHD unsteady free convective walterns memory flow with constant suction and heat sink. [37] R.M.
Mishra, G.C.Dash premeditated mass and heat transfer effect on MHD flow of a visco-elastic fluid through
porous medium with oscillatory suction and heat source. [38] Reddy et al. studied combined effect of heat
absorption and MHD on convective Rivlin-Erichsen flow past a semi-infinite vertical porous plate with variable
temperature and suction. [39] M. Umamaheswar et al. studied combined radiation and ohmic heating effects on
MHD free Convective visco-elastic fluid flow past a porous plate with viscous dissipation. [40] B.Seshaiah et
al. studied the effects of chemical reaction and radiation on unsteady MHD free convective fluid flow embedded
in a porous medium with time-depended suction with temperature gradient heat source. [41] M.C.raju et al.
studied unsteady MHD free convection and chemically reactive flow past an infinite vertical porous plate. [42]
B. Vidyasagar et al. Observed unsteady MHD free convection boundary layer flow of radiation absorbing
Kuvshinski fluid through porous medium. [43] M.C.Rajuet al. radiation absorption effect on MHD free
convection chemically reacting visco-elastic fluid past an oscillatory vertical porous plate in slip regime.MHD
three dimensional Couette flow past an exponentially accelerated vertical plate with variable temperature and
concentration in the presence of Soret and Dufour effects by Umamaheswar et al. [46].
In the past few years, there are several papers on MHD flows and chemical reactions i.e., combined
effect of heat absorption and MHD on convective Rivin – Erichsen flow past a semi-infinite vertical porous
plate with variable temperature and suction, combined radiation and ohmic heating effects on MHD free
convective visco-elastic fluid flow past a porous plate with viscous dissipation, effects of chemical reaction and
radiation on unsteady MHD free convective fluid flow embedded in porous medium with time-depended suction
with temperature gradient heat source, MHD free convection and chemical reactive flow past an infinite vertical
porous plate, Unsteady MHD free convection boundary layer flow of radiation absorbing kuvshinski fluid
through a porous medium, radiation absorption effect on MHD free convection chemically reacting visco-
elastic fluid past an oscillatory vertical porous plate in slip regime, for example papers by B. Umamaheswar et
al. [39], B. Sesaiah et al. [40], M.C.Raju et al. [41], Vidyasagar et al. [42], Raju et al.[43] and Rao et al [45] .
However, none of these papers discussed the effects of the MHD convective heat transfer flow of kuvshinski
fluid past an infinite moving plate embedded in a porous medium with radiation temperature dependent heat
source and variable suction. Thus, the aim of the present paper is to investigate the MHD convective heat
transfer flow of kuvshinski fluid past an infinite moving plated embedded in a porous medium with radiation
temperature dependent heat source and variable suction.
II. Mathematical formulation Two dimensional unsteady flows of a laminar, conducting and heat generation / absorption fluid past a
semi-infinite vertical moving porous plate embedded in a uniform porous medium and subject to a uniform
magnetic field in the presence of a pressure gradient have been considered with free convection and thermal
radiation effects. According to the coordinate system the *x -axis is taken along the porous plate in the upward
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 68 | Page
direction and *y -axis normal to it. The fluid is assumed to be gray, absorbing-emitting but non-scattering
medium. The radiative heat flux in the *x -direction is considered negligible in comparison with that in the
*y -
direction. It is assumed that there is no applied voltage of which implies the absence of an electric field. The
transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced
magnetic field is negligible. Viscous and Darcy resistance terms are taken into account in the constant
permeability porous medium. The MHD term is derived from the order of magnitude analysis of the full Navier-
stokes equation. It is assumed here that the hole size of the porous plate is significantly larger than a
characteristics microscopic length scale of the porous medium. We regard the porous medium as an assemblage
of small identical spherical particles fixed in space. The fluid properties are assumed to be constant except that
the influence of density variation with temperature has been considered in the body-force term. Due to the
semi-infinite place surface assumption furthermore, the flow variable are function of *y and
*t only. The
governing equation for this investigation is based on the balance of linear momentum energy. Taking into
consideration the assumption made above, these equations can be written in Cartesian frame of reference, as
follows;
0*
*
y
v ---(1)
2 * *
1* *1oB u
K t
---(2) ** * 2 *
* * *0
* * *2 *
1( ) r
p p p
Q qT T Tv T T
t y C y C C y
---(3)
The boundary conditions at the wall and in the free stream are:
,**
puu **
)( **** tn
WTW eTTATT 0* yat ---(4)
),1( **
0
** tneUUu **
TT as *y ---(5)
Where TA is a constant taking value 0 or 1.
If plate is with Variable Wall Temperature (VWT) then 1TA
If plate is with Constant Wall Temperature (CWT) then 0TA
Where ** ,vu -velocity components in ,X Y directions respectively, g -Gravitational acceleration,
*t
-Time, -Kinematic coefficient of viscosity, -Electrical conductivity, -The viscosity, -Density of the
fluid, *
1 -the coefficient of Kuvshinski fluid, *T -Temperature of the fluid,
*
wT -The temperature at the plate ,
*T -The temperature of fluid in free stream, - Thermal conductivity, PC -Specific heat at constant pressure,
*
rq -Radiative heat flux, *K -Permeability parameter of the porous medium, 0U and
*n - constants. The
magnetic and viscous dissipations are neglected in the study. It is assumed that the porous plate moves with a
constant velocity, *
pu in the direction of fluid flow, and the free stream velocity *U follows the exponentially
increasing small perturbation law. In addition, it is assumed that the temperature and concentration at the wall as
well as the suction velocity are exponentially varying with time.
It is clear from equation (1) that the suction velocity at the plate surface is a function of time only. Assuming
that it takes the following exponential form:
**
0
* 1 tnAeVv ---(6)
Where A is a real positive constant, and A are small less than unity and 0V is a scale of suction
velocity which has non-zero positive constant
In the free steam, from equation (2) we get
* * * 2 ** * *
1 1* * * * * *21 1
u u p uv g
t t y t x y
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 69 | Page
*
*
*2
0*
*
*
*
U
KUBg
x
p
dt
dU ---(7)
Eliminating *
*
x
P
between equation (2) and equation (7), we obtain
2*
*2
*
*
*
*
1*
**
*
*
*
*
1 11y
u
dt
dU
ty
uv
t
u
t
)(1)( **
*
*
1*
2
0 uUtK
Bg
---(8)
By making use the equation of state
* *( ) ( )T T ---(9)
Where β is the volumetric coefficient of thermal expansion and is the density of the fluid far away the
surface. Then from equation (9) and (8) we obtain
)(11 **
*
*2
*
*
*
*
1*
**
*
*
*
*
1 2
TTg
y
u
dt
dU
ty
uv
t
u
t
)(1 **
*
*
1*
2
0 uUtK
vB
---(10)
The radiation flux on the basis of the Rosseland diffusion model for radiation heat transfer is expressed as:
*
*
*
1
**
4
3
4
y
T
kqr
---(11)
Where * and
*
1k are respectively Stefan – Boltzmann constant and the mean absorption coefficient.
We assume that the temperature difference within the flow are sufficiently small such that 4*T may be
expressed as a linear function of the temperature. This is accomplished by expanding in Taylor series about *
T
and neglecting higher order terms, thus
434 **** 34 TTTT ---(12)
By using equation (12) and (13) into equation (3) is reduced to
3 4
2 2
* ** * 2 * 2 ** * *0
* * ** *1
16( )
3p p p
Q TT T T Tv T T
t y C C C ky y
---(13)
We now introduce the dimension less variables, as follows:
**
**
2
0
*2
0
*
0
*
0
*0
*
0
*
0
*
,,
,,,,
TT
TT
V
vnn
Vtt
UUuUUUVy
yV
vv
U
uu
w
pp
} ---(14)
Then substituting from equation (14) into equations (10) and (13) and taking into account equation (6), we
obtain
2
2
11 1)1(1y
u
dt
dU
ty
uAe
t
u
t
nt
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 70 | Page
)(1 1 uUt
NGr
---(15)
2
2
3
41
1)1(
y
R
PyAe
t r
nt ---(16)
Where
0
2
0
** )(
UV
TTgG w
r
(Thermal Grashof number)
2
0
2
V
BM o
(Magnetic field parameter)
2
2
0
*
VKK (Permeability parameter)
pCV
KQ2
0
*
0
(Heat generation / absorption parameter)
*
1
3**4
kk
TR
(Thermal radiation parameter)
k
CP
p
r
(Prandtl number)
2
0
*
11
V (Visco-elastic parameter)
KMN
1 ---(17)
The corresponding boundary conditions are
,pUu 1 nt
TT A e 0yat ---(18)
,1 nteUu 0 ,T yas ---(19)
Analytical Approximate Solution
In order to reduce the above system of partial differential equations to a system of ordinary equations in
dimensionless form, we may represent the velocity and temperature as
)()()( 2
10 Oyueyuu nt ---(20)
)()()( 2
10 Oyey nt ---(21)
By substituting the above equations (20),(21) into equations (15)&(16), equating the harmonic and non-
harmonic term and neglecting the higher – order terms of O ( )( 2 ,we obtain the following pairs of equations
for ),( 0ou and ),( 11 u
orooo GNNuuu ---(22)
111111 ))(1()1)(( ro GuAnNnunnNuu ---(23)
0Pr3Pr3)43( 0 ooR ---(24)
oAnR Pr3)(3Pr3)43( 111 ---(25)
Where, the primes denote differentiation with respect to y.
The corresponding boundary conditions are
,po Uu ,01 u ,10 TA1 0yat
---(26)
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 71 | Page
,1ou ,11 u 00 01 yas ---(27)
The analytical solutions of equations (22) to (25) with satisfying boundary conditions (26) and (27) are given
by ymyL
p ePePuu 00
110 1)1(
---(28)
ymymyLyLeReReRecu 1001
32111 1
---(29)
ym
o e 0 ---(30)
ymym
T ePePA 01
001 )(
---(31)
Where
Ra 430 110 PuA p
2
4 00
2
00
0
cabbm
10
2
0
01
ALL
BR
rPb 30 000 LAAB
2
4 10
2
00
1
aabbm
10
2
0
02
Amm
CR
rPc 30 0010 PGmAPC r
100
2
00
100
ambma
bmP
11
2
1
13
Amm
BR
na 31 11 1 nnNA
Nmm
GP r
0
2
0
1
rAPb 31
01 PAGB Tr
2
4110
NL
01 PCAT
01 PAC T
2
411 1
1
AL
In view of the above solution, the velocity and temperature distribution in the boundary layer become ymyL
p ePePutyu 00
11 1)1(),(
)1( 1001
3211
ymymyLyLnt eReReRece
---(32)
ymety 0),(
])[( 01
00
ymym
T
nt ePePAe
---(33)
It is now important to calculate the physical quantities of primary interest, which are the local wall
shear stress, the local surface heat flux. Given the velocity field in the boundary layer, we can now calculate the
local wall shear stress (i.e., skin-friction) is given by
0*
**
yw
y
u ---(34)
And in dimensionless form, we obtain
0
*
yoo
wf
y
u
VUC
= 0 1 1 0( 1)PL u P Pm
1 1 1 0 2 0 3 1( )nte C L R L R m R m ---(35)
Similarly we calculate the heat transfer coefficient in terms of Nusselt number, as follows
0y
uy
N
])([ 00010 PmPAmem T
nt ---(36)
III. Results and Discussions In order to get physical insight into the problem , the velocity and temperature fields have been
discussed by assigning numerical values of Thermal radiation parameter R ,Prandtlnumbe rP, Thermal Grashof
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 72 | Page
number rG,Permeability parameter K , Magnetic field parameter M , Visco-elastic parameter 1 , heat
generation / absorption and pU in two cases.
Case i):- The plate with variable wall temperature (VWT),i.e 1TA
Fig. 1 illustrates the behavior of the velocity for different values of R . It seen that the velocity
increases with the increasing R . Fig.2 presents the variation of velocity component for various values of rP .
The result shows that the effect increasing values rP in a decreasing velocity. The velocity profile against y
for different values of rG described in fig.3 it is observed that an increase in rG leads to a rise the velocity.
Fig 1: Velocity profile against y for different values of R at 0.1TA
;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Fig 2: Velocity profile against y for different values of R at 0.1TA
;0.1R 0.1A ;0.5Gr ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 73 | Page
Fig 3: Velocity profile against y for different values of Gr at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
The effect of the K on the velocity profile has been shown in Fig. 4. It is observed that the velocity
increases as K increases. Fig. 5 shows that the velocity profiles for different values of M . It is noticed that
velocity decreases with the increase of M . Fig.6 illustrates the variation of velocity distribution across the
boundary layer for several values of plate moving velocity in the direction of fluid flow. The velocity increases
with the increase of pU . From Fig.7 it is observed that velocity profile increases with increasing 1 . It is
observed from Fig. 8 that the velocity increases it increasing the .
Fig 4: Velocity profile against y for different values of K at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.5Gr 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 74 | Page
Fig 5: Velocity profile against y for different values of M at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Fig 6: Velocity profile against y for different values of Up at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 75 | Page
Fig 7: Velocity profile against y for different values of 1 at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
1.0;Up ;01.0 ;02.0 ;2.0n 0.1t
Fig 8: Velocity profile against y for different values of at 0.1TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
1.0;Up 1 0.01; ;02.0 ;2.0n 0.1t
The effect of R on temperature profiles are shown in Fig.9. It is observed that the temperature
increases with the increase of R . Fig.10 shows the effects of rP on temperature profile. It is noticed that
temperature decreases with the increase in rP . It is observed in Fig. no.11 that temperature increases with
increasing .
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 76 | Page
Fig 9: Temperature profile against y for different values of R at 1TA
;71.0Pr 0.1A ;01.0 ;02.0 ;1.0n 0.1t
Fig 10: Temperature profile against y for different values of Pr at 1TA
;0.1R 0.1A ;01.0 ;02.0 ;1.0n 0.1t
Fig 11: Temperature profile against y for different values of at 1TA
;71.0Pr ;0.1R 0.1A ;02.0 ;1.0n 0.1t
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 77 | Page
Case ii):- Plate is with Constant wall temperature (CWT),i.e., 0TA
The velocity profile against y for different values of R described in fig. 12. It is observed that an
increase in R leads to a rise the velocity. Fig.13 shows the effects of rP on velocity profile. It is noticed that
velocity decreases with the increase in rP .The effect of the rG on the velocity profile has been shown in Fig.
14. It is observed that the velocity increases with increasing the rG .
Fig 12: Velocity profile against y for different values of R at 0TA
;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Fig 13: Velocity profile against y for different values of Pr at 0TA
;0.1R 0.1A ;0.5Gr ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
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Fig 14: Velocity profile against y for different values of Gr at 0TA
;0.1R ;71.0Pr 0.1A ;0.1K 0.1M
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
The effect of K on velocity profiles are shown in Fig.15. It is observed that the velocity increases
with the increase of K .Fig.16 shows that the velocity profile decreases with increasing M . Fig.17 illustrates the
variation of velocity distribution across the boundary layer for several values of plate moving velocity in the
direction of fluid flow. The velocity increases with the increase of pU . Fig.18 shows that the velocity increases
with increase on 1 Fig.19 illustrate that the velocity increases with increase on
Fig 15: Velocity profile against y for different values of K at 0TA
;0.1R ;71.0Pr 0.1A ;0.5Gr 0.1M
;0.1Up ;01.01 ;01.0 ;2.0n 0.1t
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
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Fig 16: Velocity profile against y for different values of M at 0TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K
;0.1Up ;01.01 ;01.0 ;02.0 ;2.0n 0.1t
Fig 17: Velocity profile against y for different values of Up at 0TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
;01.01 ;01.0 ;02.0 ;2.0n 0.1t
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Fig 18: Velocity profile against y for different values of 1 at 0TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
1.0;Up ;01.0 ;02.0 ;2.0n 0.1t
Fig 19: Velocity profile against y for different values of at 0TA
;0.1R ;71.0Pr 0.1A ;0.5Gr ;0.1K 0.1M
1.0;Up 1 0.01; ;02.0 ;2.0n 0.1t
Fig.20 presents the variation of temperature component for various values of R . The result shows that
the effect of increasing values R results in an increasing temperature. Fig. 21 illustrates the behavior of the
temperature for different values of rP . It is seen that the temperature decreases with the increasing rP .Fig.22
shows that the temperature increases with increase on .
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
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Fig 20: Temperature profile against y for different values of R at 0TA
;0.1R ;71.0Pr 0.1A ;01.0 ;02.0 ;1.0n 0.1t
Fig 21: Temperature profile against y for different values of Pr at 0TA
;0.1R 0.1A ;01.0 ;02.0 ;1.0n 0.1t
Fig 22: Temperature profile against y for different values of at 0TA
;71.0Pr ;0.1R 0.1A ;02.0 ;1.0n 0.1t
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From table 1&2 it is noticed that an increases in 1, ,rP increases the co-efficient of skin-friction in both cases
1TA and 0TA . We also observed that ,R ,rG ,K M and Up increases the co-efficient of skin-friction
decreases in both cases 1TA and 0TA .
Table 3&4 shows that an increase in Pr decreases the rate of heat transfer Nu in both cases
1TA and 0TA . And we noticed that an increase in ,R increases the rate of heat transfer in both cases
1TA and 0TA .
Table 1: Co-efficient of skin-friction at variable wall temperature (VWT),i.e 1TA
R Pr Gr K M Up 1
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -1.7200
5.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -1.8209
10.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -2.0494
15.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -2.3835
1.0 0.8 5.0 1.0 1.0 1.0 0.01 0.01 -1.5850
1.0 0.9 5.0 1.0 1.0 1.0 0.01 0.01 -1.4584
1.0 1.0 5.0 1.0 1.0 1.0 0.01 0.01 -1.3512
1.0 0.71 7.0 1.0 1.0 1.0 0.01 0.01 -2.3879
1.0 0.71 9.0 1.0 1.0 1.0 0.01 0.01 -3.0557
1.0 0.71 12.0 1.0 1.0 1.0 0.01 0.01 -4.0579
1.0 0.71 5.0 2.0 1.0 1.0 0.01 0.01 -1.8181
1.0 0.71 5.0 3.0 1.0 1.0 0.01 0.01 -1.8570
1.0 0.71 5.0 4.0 1.0 1.0 0.01 0.01 -1.8779
1.0 0.71 5.0 1.0 2.0 1.0 0.01 0.01 -1.5773
1.0 0.71 5.0 1.0 3.0 1.0 0.01 0.01 -1.4752
1.0 0.71 5.0 1.0 4.0 1.0 0.01 0.01 -1.3966
1.0 0.71 5.0 1.0 1.0 1.2 0.01 0.01 -1.3168
1.0 0.71 5.0 1.0. 1.0 1.4 0.01 0.01 -0.9137
1.0 0.71 5.0 1.0 1.0 1.6 0.01 0.01 -0.5105
1.0 0.71 5.0 1.0 1.0 1.0 1.0 0.01 -1.7225
1.0 0.71 5.0 1.0 1.0 1.0 3.0 0.01 -1.7270
1.0 0.71 5.0 1.0 1.0 1.0 5.0 0.01 -1.7311
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.03 -1.8142
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.06 -2.0479
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.08 -2.4876
Table 2: Co-efficient of skin-frictionatat constant wall temperature (CWT),i.e 0TA
R Pr Gr K M Up 1
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -1.6922
5.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -1.8002
10.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -2.0325
15.0 0.71 5.0 1.0 1.0 1.0 0.01 0.01 -2.3687
1.0 0.8 5.0 1.0 1.0 1.0 0.01 0.01 -1.5584
1.0 0.9 5.0 1.0 1.0 1.0 0.01 0.01 -1.4331
1.0 1.0 5.0 1.0 1.0 1.0 0.01 0.01 -1.3271
1.0 0.71 7.0 1.0 1.0 1.0 0.01 0.01 -2.3489
1.0 0.71 9.0 1.0 1.0 1.0 0.01 0.01 -3.0057
1.0 0.71 12.0 1.0 1.0 1.0 0.01 0.01 -3.9907
1.0 0.71 5.0 2.0 1.0 1.0 0.01 0.01 -1.7892
1.0 0.71 5.0 3.0 1.0 1.0 0.01 0.01 -1.8277
1.0 0.71 5.0 4.0 1.0 1.0 0.01 0.01 -1.8484
1.0 0.71 5.0 1.0 2.0 1.0 0.01 0.01 -1.5513
1.0 0.71 5.0 1.0 3.0 1.0 0.01 0.01 -1.4504
1.0 0.71 5.0 1.0 4.0 1.0 0.01 0.01 -1.3730
1.0 0.71 5.0 1.0 1.0 1.2 0.01 0.01 -1.2891
1.0 0.71 5.0 1.0. 1.0 1.4 0.01 0.01 -0.8859
1.0 0.71 5.0 1.0 1.0 1.6 0.01 0.01 -0.4827
1.0 0.71 5.0 1.0 1.0 1.0 1.0 0.01 -1.6955
1.0 0.71 5.0 1.0 1.0 1.0 3.0 0.01 -1.7014
1.0 0.71 5.0 1.0 1.0 1.0 5.0 0.01 -1.7066
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.03 -1.7850
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.06 -2.0185
1.0 0.71 5.0 1.0 1.0 1.0 0.01 0.08 -2.4574
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Table 3:Nusselt Number at variable wall temperature (VWT),i.e 1TA
R Pr Nu
1.0 0.71 0.01 -2.1166
5.0 0.71 0.01 -1.9489
10.0 0.71 0.01 -1.6324
15.0 0.71 0.01 -1.1802
1.0 0.8 0.01 -2.3920
1.0 0.9 0.01 -2.6980
1.0 1.0 0.01 -3.0041
1.0 0.71 0.03 -1.9497
1.0 0.71 0.05 -1.7405
1.0 0.71 0.07 -1.4122
Table 4: Nusselt Number at constant wall temperature (CWT),i.e 0TA
R Pr Nu
1.0 0.71 0.01 -2.0546
5.0 0.71 0.01 -1.8655
10.0 0.71 0.01 -1.5300
15.0 0.71 0.01 -1.0625
1.0 0.8 0.01 -2.3251
1.0 0.9 0.01 -2.6256
1.0 1.0 0.01 -2.9260
1.0 0.71 0.03 -1.8905
1.0 0.71 0.05 -1.6843
1.0 0.71 0.07 -1.3593
IV. Conclusion
We have discussed the effects of the MHD convective heat transfer flow of Kuvshinski fluid past an
infinite moving plate embedded in a porous medium with radiation temperature dependent heat source and
variable suction. Thus, the aim of the present paper is to investigate the MHD convective heat transfer flow on
Kuvshinski fluid past an infinite moving plate embedded in a porous medium with temperature dependent heat
source and variable suction. The following are the concluding remarks.
a. When the plate is maintained at variable temperature, the velocity is observed to increases with the increasing
values of 1, , , , ,Gr k Up R where as it has reverser effect in the case of rP and M . Temperature boundary
layer increased with the increase in ,R but decrease for increasing values of Pr .
b. When the plate is maintained at constant wall temperature, the velocity is observed to increase with the
increasing values of 1, , , , ,Gr k Up R where as it has reverser effect in the case of rP and M . Temperature
boundary layer increased with the increase in ,R but decrease for increasing values of Pr .
References [1]. Pohlhauses H. Der Wrmeaustrausch Zwischenfesten Korpern and Flussingkeitenmit Kleiner Reibung under KleinerWarmeleitung
ZAMM ;1:115-121, 1921.
[2]. Ostrach.S ,“An analysis of laminar free convection flow and heat transfer along a flat plate parallel to the direction of the generating body force “NACA rep. 1111, 1953.
[3]. P.G .Saffirman, “On the stability of laminar flow of a dusty gas”, Journal of fluid mechanics, Issue 1, volume 13, pp 120-129, 1962.
[4]. Gill WN, Del Casal E, Zeh DW. Binary, “Diffusion and heat transfer in laminar free convection boundary layers on a vertical
plate”. International Journal Heat Mass Transfer, Volume 8, pp 1131-1151, 1965.
[5]. Takhar H.S, Golra R.S.R. and SoundalgekarV.M,“Radiation effects on MHD free convection flow of a radiating gas past a semi–
infinite plate”. International Journal of Numerical Methods Heat Fluid Flow, vol 6 (20, pp. 77-83, 1966. [6]. V.M.Soundalegkar,“Approximate solutions for the two-diffusional flow of an impressible, viscous fluid past an infinite porous
vertical plate with constant suction velocity normal to the plate “ . Proc. Roy.Soc. London, A333, 1973.
[7]. V.M.Soundalgker, H.S. Takkar, “Magnetohydrodynamics (MHD) generator, plasma studies,nuclear reactors, geothermal energy extractions and the boundary layer control in the field of aerodynamics”. AIAA, J.15, 457, 1977.
[8]. Raptis and Kafousians ,“Influence of a magnetic field upon the study free convection flow through a porous medium bounded by an infinite vertical plate with constant suction velocity”. International Journal Energy Res. 6.241, 1982.
[9]. A.A.Raptis,“Time-varying two-dimensional natural convective heat transfer on an incompressible, electrically-conductive viscous
fluid via a highly porous medium bounded by an infinite vertical porous plate”. International J.Energy Res. 10, 97, 1986. [10]. H.T.Chen, C.K.Chen “Free convection of non-Newtonian Fluid Flow along a vertical plate embedded in a porous medium”, Trans,
ASME, J.Heat transfer, Vol, 110, pp.257-260, 1988.
[11]. KeeSooHan, Hyung Jin Sung and MyungKyoonChung,“Analysis of heat transfer in a pipe carrying two – phase gas particle suspension”, International Journal of Heat and Mass transfer, volume 34, Issue1, p.69-78, 1991.
[12]. Sharma PR and PankajMathur,“Steady laminar free convection flow of an electrically conducting fluid along a porous hot vertical
plate in the presence of Heat source/sink”, Indian Journal Pure appl.math, 26(11), pp. 1125-1134, 1995.
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 84 | Page
[13]. Takhar H.S, Golra R.S.R. and Soundalgekar V.M., “Radiation effects on MHD free convection flow of a radiating gas past a semi-
infinite vertical plate”,Int.J. Numerical Methods Heat Fluid Flow, 6, pp. 77-83, 1996.
[14]. M.A.Mansour,“Forced convection-radiation interaction heat transfer in boundary layer over a flat plate submerged in a porous medium”,Applied Mechanical Engineering , Vol.3, pp.405-413, 1997.
[15]. Helmy A.K “MHD unsteady free convection flows past a vertical porous plate”, ZAMM Vol, 78(4), pp.225-270, 1998.
[16]. Edmundo M., Nune and Mohammad H.N. Naragh,“Numerical model for radiative heat transfer analysis in arbitrary shaped axisymmetric enclosures with gases media”, Numerical Heat Transfer, vol.33. issue 5, pp. 495-513, 1998.
[17]. A.Raptis,“Radiation and free convection flow through a porous medium”. Int. Commun. Heat Mass transfer,Vol.2.pp. 289-295,
1998. [18]. M.A.Hossain, M.AAlim, andD.A.Ress, “The effect of radiation in free convection from a porous vertical plate”,Int J. heat and
Mass transfer, vol. 42, pp. 131, 1999.
[19]. M.A. El.Hakiem, “MHD oscillatory flow on free convection-radiation through a porous medium with constant suction velocity”. J. of Magnetism and magnetic materials, vol. 220 pp. 271 – 276, 2000.
[20]. Acharya M. Dash GC and Singh LP,“Magnetic field effects on the free convection and mass transfe flow through porous medium
with constant suction and constant heat flux”, Indian J. Pure appl. Math., 31 (1),99 1-18, 2000. [21]. Kim Youn J. “Unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction”, Int.
J.Eng. Sci. 38, pp. 833-845, 2000.
[22]. E.M.AboEldahab and M.S.E.Gendy , “Radiation effects on convective heat transfer in an electrically conducting fluid at a stretching surface with variable viscosity and uniform free stream”. Physicascripta, vol 62, pp. 321-325, 2000.
[23]. E.M.AboEldahab and M.S.E.Gendy.“Convective heat transfer past a continuously moving plate embedded in a non-darcian porous
medium in the presence of a magnetic field”, Canadian Journal of Physics, Vol 97, pp. 1031-1038, 2001. [24]. Sahoo PK, Datta N, and Biswal S. “ Magnetohydrodynamic unsteady free convection flow past an infinite vertical plate with
constant suction and heat sink”, Indian J. Pure appl. Math., 34(1), pp 145-155, 2003.
[25]. Isreal-cookey.C,Ogulu.A,Omubo-pepple .V.B. “Influence of viscous dissipation and radiation on unsteady MHD free –convection flow past an infinite heated vertical plate in a porous medium with time dependent suction”, International Journal of Heat and Mass
Transfer Vol. 46, pp. 2305-2311, 2003.
[26]. Raptis, C.Perkikis and H.S.Takhar, “Effect of thermal radiation on MHD flow”, Applied Mathematics and computation, vol. 153, pp. 645-649, 2004.
[27]. R.K.Varshney and Ram Prakash “MHD free convection flow of a visco-elastic dusty gas through a porous medium induced by the
motion of a semi-infinite flat plate moving with velocity decreasing exponentially with time”,ActaCienciaIndica (India) vol. XXX M, No.2, pp. 467-473, 2004.
[28]. R. Muthukumaraswamy and G. Kumar Senthil,“Heat and Mass transfer effect on moving vertical plate in the presence of thermal
radiation”, Theoretical applied Mechanics, Vol.3, pp. 35, 2004. [29]. Chamkaali J. “Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat
absorption”, Int. J. Engg. Sci. 42, pp 217-230, 2004.
[30]. S.Shateyi, P.Sibanda, and S.S. Motsa, “Magnetohyderodynamics flow past a vertical plate with radiative heat transfer”,Journalof
Heat Transfer, Vo, 129, pp. 1708-1714, 2007.
[31]. Ibrahim.F.S., Elaiw. A.M., BakrA.A.,“Effect of the chemical reaction and radiation absorption on the unsteady MHD free
convection flow past a semi-infinite vertical permeable moving plate with het source and suction”, Comm.Nonlinear Sci. Number.
Simul 13 (6), 1056-1066, 2008. [32]. P.M Patil, P.S.Kulkarni,“Effects of chemical reaction on free convection flow of a polar fluid through a porous medium in the
presence of Internal Heat Generation”, Internal Journal of thermal science, Vol. 47, No. 8 pp. 1043-1054, 2008.
[33]. O.A. Beg, J. Zueco, R. Bhargava and H. S. Jakhar, “Magneto hydrodynamic convection flow from a sphere to a non-darcian porous medium with heat generation or absorption effects network simulation”, International Journal of Themal Science, vol. 48, pp. 1813-
1830, 2009.
[34]. Kumar .H,“Radiative heat transfer with hydromagenatic flow and viscous dissipation over a stretching surface in the presence of variable heat flux”, Thermal science, vol. 13 pp. 163-169, 2009.
[35]. O.D.Makinde and T. Chinyoka, “Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative
heat and convective cooling”, Communication in Nonlinear science and numerical simulations, vol. 15, pp. 3919-3930, 2010. [36]. J. Gireesh Kumar, P.V.Satynarayana,“Mass transfer effect on MHD unsteady free convective walterns memory flow with constant
suction and heat sink”,Int.J. Appl. Math. Mech. 9 (7), pp. 66-94, 2013.
[37]. R.M. Mishra, G.C.Dash, “Mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source”,IJHMT Vo. 57 pp. 433438, 2013.
[38]. ReddyA.N, M.C.Raju, G.S.S.Raju, “Combined effect of heat absorption and MHD on convective Rivlin-Erichsen flow past a semi-
infinite vertical porous plate with variable temperature and suction”,Ain Shams Eng J. 2014. [39]. M. Umamaheswara, S,V.K.Varma, M.C.Raju, “Combined Radiation and OhmicHeatingEffects on MHD free Convective visco-
elastic fluid flow past a porous plate with viscous dissipation”, Internal of current engineering and technology, Vol.3. no.5, pp.
1636-1640, 2013. [40]. B.Seshaiah, S.V.K. Varma, M.C.raju, “The effects of chemical reaction and radiation on unsteady MHD free convective fluid flow
embedded in a porous medium with time-depended suction with temperature gradient heat source”, International Journal of
Scientific Knowledge, Vol. 3, NO.2 pp. 13-24, 2013. [41]. M.C.Raju, S.V.K.Varma, and R.R.K.Rao,”Unsteady MHD free convection and chemically reactive flow past an infinite vertical
porous plate”, Journal of Future Engineering and Technology, Vol. 8. pp. 35-40, 2013.
[42]. B. Vidyasagar, M.C.Raju, S.V.K. Varma, S.Venkataramana,”Unsteady MHD free convection boundary layer flow of radiation absorbing Kuvshinski fluid through porous medium”, Review of Advance in physics Theories and Applications, 1(3): pp.48-62,
2014.
[43]. M.C.Raju, B.Vidyasagar, S.V.K.Varma, S.Venkataramana, “Radiation absorption effect on MHD free convection chemically reacting visco-elastic fluid past an oscillatory vertical porous plate in slip regime”, International Journal of Engineering and
Computer science, Vol.3 (9), pp. 8212-8221, 2014.
[44]. Ravikumar V, Raju M.C, Raju G.S.S.,MHD three dimensional Couette flow past a porous plate with heat transfer, IOSR Jour.Maths., Vol. 1, no.3,pp.3-9,2010.
Mhd Convective Heat Transfer Flow Of Kuvshinski Fluid Past An Infinite Moving Plate Embedded ..
DOI: 10.9790/5728-1203066685 www.iosrjournals.org 85 | Page
[45]. B.M.Rao, G.V.Reddy, M.C.Raju., S.V.K,Varma, MHD transient free convection and chemically reactive flow past a porous vertical
plate with radiation and temperature gradient dependent heat source in slip flow regime, IOSR Journal of applied physics, Vol. 3(6),
2013,22-32. [46]. M.Umamaheswar, S.V.K.Varma, M.C.Raju, Numerical study of Magneto-Convective and radiation abosorption fluid flow past an
exponentially accelerated vertical porous plate with variable temperature and concentration in the presence of Soret and Dufour
effects, IOSR Journal of Mathematics, Volume12, Issue 2 Ver. I (Mar.-Apr.2016), pp 109-120. DOI:10.9790/5728-1221109120.